Effects of near-neighbor correlations on the diffuse scattering from a one-dimensional paracrystal

Abstract

The one-dimensional paracrystal model is generalized by folding the lattice sites with objects whose scattering lengths or sizes and separation display a spatial correlation from cell to cell. A general theory to calculate the diffuse scattering and the scattering-length autocorrelation function is developed. The investigated models of coupling along the paracrystalline chain are the correlations between (i) the sizes of the scatterers, (ii) the sizes of scatterers and their separations, and (iii) the sizes of scatterers and the fluctuation of their separation distances. In the first case (i), the size of a scatterer is, on average, linked to that of its neighbors. As a result, a continuous transition from the total lack of size correlation (known as decoupling approximation or DA) to the scattering from monodisperse domains (local monodisperse approximation or LMA) is obtained. In the second case of correlation (ii), the mean interobject distance is assumed to depend on the respective sizes of nearest neighbors. Depending on the introduced correlation parameter, aggregation or hard-core-type effects can be accounted for. Surprisingly, in some cases, it is possible to find a peak in the scattering curve without any structure in the total interference function. The size-separation correlations may also dramatically reduce the scattering intensity close to the origin compared to the completely uncorrelated case. The last model (iii) foresees a coupling between the sizes of neighboring objects and the variance of the separation between neighbors. Within this model, on average along the chain, the fluctuations of distances between scatterers become dependent on the respective sizes of neighbors, while the mean distance between objects remains constant.

Probing surface and interface morphology with Grazing Incidence Small Angle X-Ray Scattering

“…A smooth transition between DA and LMA is obtained by gradually increasing the size correlation between neighbors. Leroy et al [235] illustrated this result in the framework of particles aligned along one dimension (see Fig. 54).…”

“…The Local Monodisperse Approximation introduced by Pedersen [234] assumes that the particle collection is made of monodisperse domains which size are larger than the coherence length of the X-ray beam. At variance to the DA, a nearly perfect correlation between the size and shape of neighboring particles is assumed [235]. As the monodisperse domains interfere incoherently, the LMA cross-section reads:…”

“…The Size-Spacing Correlation approximation (SSCA) [235,155,81] is a one-dimensional analytical modeling of scattering from correlated particles. It gives a fairly good illustration of the problem at hand although its applicability to actual measurements of SAXS [81] is questionable owing to its 1D character.…”

Grazing incidence X-ray study of Ge-nanoparticle formation in (Ge:SiO2)/SiO2 multilayers

“…The SSCA model relies on hypothesis of correlation between the particle's diameter and the distance to its first neighbor. In short, this model is based on one-dimensional paracrystal model in which the spacing probability between two neighboring particles having radius R n − 1 and R n is chosen such that their average separation depends linearly on their sizes and is equal to d = L || + κ(R n − 1 + R n − D || ) [16] (L || is the mean distance between particles irrespective of their sizes, D || the mean particle diameter and κ a size-spacing correlation factor). In this model, for larger κ, the separation of two neighboring particles becomes more dependent on their sizes: smaller particles become closer to each other while larger particles are further apart.…”

“…For this T a range, 1D lateral profiles were fitted with the IsGISAXS software [15], using size-spacing correlation approximation (SSCA) [16] for determination of the in-plane morphological parameters. The SSCA model relies on hypothesis of correlation between the particle's diameter and the distance to its first neighbor.…”